reserve r1,r2 for Real;
reserve n,i,i1,i2,j for Nat;
reserve D for non empty set;
reserve f for FinSequence of D;

theorem Th8:
  for f being FinSequence of TOP-REAL 2, p being Point of TOP-REAL
  2 st p in L~f holds 1<=Index(p,f) & Index(p,f) < len f
proof
  let f be FinSequence of TOP-REAL 2, p be Point of TOP-REAL 2;
  assume p in L~f;
  then consider S being non empty Subset of NAT such that
A1: Index(p,f) = min S and
A2: S = { i: p in LSeg(f,i) } by Def1;
  Index(p,f) in S by A1,XXREAL_2:def 7;
  then
A3: ex i st i = Index(p,f) & p in LSeg(f,i) by A2;
  hence 1 <= Index(p,f) by TOPREAL1:def 3;
  Index(p,f) + 1 <= len f by A3,TOPREAL1:def 3;
  hence thesis by NAT_1:13;
end;
